Philosophy of Set Theory
|
|
- Giles Blake
- 5 years ago
- Views:
Transcription
1 Philosophy of Set Theory A Pragmatistic View Yang Rui Zhi Department of Philosophy Peking University September 15, 2011 Yang Rui Zhi (PKU) Philosophy of Set Theory 15 Sep / 15
2 Outline 1 A Pragmatistic View on the Philosophy of Mathematics 2 Universe and Multiverse Multiverse Axioms Yang Rui Zhi (PKU) Philosophy of Set Theory 15 Sep / 15
3 Pragmatism Pragmatism is introduced to solve dilemmas Basic Rule of Pragmatism The meaning of concept, statement or opinion should be clarified by considering their practical consequences Remark: Epistemological puzzle: experience The way to Ethical judgement Yang Rui Zhi (PKU) Philosophy of Set Theory 15 Sep / 15
4 Mathematics Provide Knowledge Question If mathematical truths are analytic, does mathematics still gives knowledge? If mathematics gives knowledge, what form the knowledge is presented in? My Answer: The knowledge is given in the following form: There exists (constructively) a sequence witnessing the fact: Λ L φ Yang Rui Zhi (PKU) Philosophy of Set Theory 15 Sep / 15
5 Absoluteness of Mathematical Achievements A mathematical achievement will never fade away even the axioms are no longer recognized, Russell s Paradox even the underground logic is discarded, Intuitionistic Logic even the subject itself is out of the stage Gödel s coding The structure of practical consequences is atomic Yang Rui Zhi (PKU) Philosophy of Set Theory 15 Sep / 15
6 Philosophy gives Programs Frege s Program Motivation: Logicism; Consequences: formalization of predicate logic and axiomatization of set theory Hilbert s Program Motivation: Formalism and Finitism; Consequences: proof theory, Gödel s Incompleteness theorem Other program based on constructivism intuitionistic analysis, finitistic mathematics Yang Rui Zhi (PKU) Philosophy of Set Theory 15 Sep / 15
7 Gödel s Program The phenomenons of incompleteness Realism Large cardinals The well-ordering of large cardinals What large cardinals can do: consistency, V L, PD, etc What large cardinals cannot do: CH Inner model program Bad news: It is extremely hard to move up Good news: Ultimate L Yang Rui Zhi (PKU) Philosophy of Set Theory 15 Sep / 15
8 Friedman s Program Find simple (Π 0 2 or even Π0 1 ) and natural (non-metamathematical, eg consistency) arithmetical statements that require strong systems to settle Yang Rui Zhi (PKU) Philosophy of Set Theory 15 Sep / 15
9 Philosophy Predicts Definition (Reinhardt 1967) κ is a Reinhardt cardinal if there exists an non-trivial elementary embedding j : V V such that the critical point of j is κ Theorem (Kunen 1971) Ạssuming AC There is no Reinhardt cardinal Open Problem Ẉhether ZF + Reinhardt cardinal is consistent Realism predicts it is inconsistent Yang Rui Zhi (PKU) Philosophy of Set Theory 15 Sep / 15
10 Philosophy Sets Barriers Russell s ramified type theory and Gödel s L What if the V = Ultimate L? Yang Rui Zhi (PKU) Philosophy of Set Theory 15 Sep / 15
11 The Universe View Universe and Multiverse Multiverse Axioms The universe view is the traditional realistic view on set theory It holds that the universe consists of all sets, and we are supposed to explore in the universe and uncover the the truths of it ZF, AC, and even large cardinals are considered as our discovery of the universe Mathematicians observation of the universe can be wrong as physicists can be wrong of the physical universe Yang Rui Zhi (PKU) Philosophy of Set Theory 15 Sep / 15
12 The Multiverse View Universe and Multiverse Multiverse Axioms The Multiverse View Ṭhere are many set-theoretic universes In other words, there are numerous distinct concepts of set, no just one absolute concept of set The Multiverse view is claimed to be a realism, a second order realism about universes Yang Rui Zhi (PKU) Philosophy of Set Theory 15 Sep / 15
13 Arguments Against Universe Universe and Multiverse Multiverse Axioms Mathematical History: Irrational number Complex number Non-Euclidean geometry The fact that we are so familiar with living in different universes of set theory makes it impossible to settle problems like CH as the universe view hopes Yang Rui Zhi (PKU) Philosophy of Set Theory 15 Sep / 15
14 Universe and Multiverse Multiverse Axioms Multiverse Axioms (Hamkins, Gitman) Inner models are exists as a universe Forcing extensions are exists as a universe For each universe V, there exists a taller universe W such that V is an initial segment of W Every universe is countable from the perspective of another universe Every universe is ill-founded from the perspective of another universe For every universe V and every embedding j : V M, there exists a universe W and an embedding h such that j is the iterate of h: W h V j M Every universe V is a countable transitive model in another universe satisfying V = L Yang Rui Zhi (PKU) Philosophy of Set Theory 15 Sep / 15
15 Universe and Multiverse Multiverse Axioms The Consistency of Multiverse Axioms Theorem Gitman, Hamkins If ZFC is consistent, then the Multiverse Axioms are consistent Actually, if there exists a model of ZFC, then the collection of countable computably-saturated models of ZFC satisfies all the multiverse axioms Trick in the proof Yang Rui Zhi (PKU) Philosophy of Set Theory 15 Sep / 15
How Philosophy Impacts on Mathematics
.. How Philosophy Impacts on Mathematics Yang Rui Zhi Department of Philosophy Peking University Fudan University March 20, 2012 Yang Rui Zhi (PKU) Philosophical Impacts on Mathematics 20 Mar. 2012 1 /
More informationThe triple helix. John R. Steel University of California, Berkeley. October 2010
The triple helix John R. Steel University of California, Berkeley October 2010 Three staircases Plan: I. The interpretability hierarchy. II. The vision of ultimate K. III. The triple helix. IV. Some locator
More informationVictoria Gitman and Thomas Johnstone. New York City College of Technology, CUNY
Gödel s Proof Victoria Gitman and Thomas Johnstone New York City College of Technology, CUNY vgitman@nylogic.org http://websupport1.citytech.cuny.edu/faculty/vgitman tjohnstone@citytech.cuny.edu March
More informationA Natural Model of the Multiverse Axioms
Notre Dame Journal of Formal Logic Volume 51, Number 4, 2010 A Natural Model of the Multiverse Axioms Victoria Gitman and Joel David Hamkins Abstract If ZFC is consistent, then the collection of countable
More informationClass 15: Hilbert and Gödel
Philosophy 405: Knowledge, Truth and Mathematics Spring 2008 M, W: 1-2:15pm Hamilton College Russell Marcus rmarcus1@hamilton.edu I. Hilbert s programme Class 15: Hilbert and Gödel We have seen four different
More informationarxiv: v1 [math.lo] 7 Dec 2017
CANONICAL TRUTH MERLIN CARL AND PHILIPP SCHLICHT arxiv:1712.02566v1 [math.lo] 7 Dec 2017 Abstract. We introduce and study a notion of canonical set theoretical truth, which means truth in a transitive
More informationAbsolutely ordinal definable sets
Absolutely ordinal definable sets John R. Steel University of California, Berkeley May 2017 References: (1) Gödel s program, in Interpreting Gödel, Juliette Kennedy ed., Cambridge Univ. Press 2014. (2)
More informationThe set-theoretic multiverse: a model-theoretic philosophy of set theory
The set-theoretic multiverse: a model-theoretic philosophy of set theory Joel David Hamkins The City University of New York The College of Staten Island of CUNY & The CUNY Graduate Center New York City
More informationINCOMPLETENESS I by Harvey M. Friedman Distinguished University Professor Mathematics, Philosophy, Computer Science Ohio State University Invitation
INCOMPLETENESS I by Harvey M. Friedman Distinguished University Professor Mathematics, Philosophy, Computer Science Ohio State University Invitation to Mathematics Series Department of Mathematics Ohio
More informationGödel s Programm and Ultimate L
Gödel s Programm and Ultimate L Fudan University National University of Singapore, September 9, 2017 Outline of Topics 1 CT s Problem 2 Gödel s Program 3 Ultimate L 4 Conclusion Remark Outline CT s Problem
More informationProof Theory and Subsystems of Second-Order Arithmetic
Proof Theory and Subsystems of Second-Order Arithmetic 1. Background and Motivation Why use proof theory to study theories of arithmetic? 2. Conservation Results Showing that if a theory T 1 proves ϕ,
More informationSemantic methods in proof theory. Jeremy Avigad. Department of Philosophy. Carnegie Mellon University.
Semantic methods in proof theory Jeremy Avigad Department of Philosophy Carnegie Mellon University avigad@cmu.edu http://macduff.andrew.cmu.edu 1 Proof theory Hilbert s goal: Justify classical mathematics.
More informationPhilosophy of Mathematics Intuitionism
Philosophy of Mathematics Intuitionism Owen Griffiths oeg21@cam.ac.uk St John s College, Cambridge 01/12/15 Classical mathematics Consider the Pythagorean argument that 2 is irrational: 1. Assume that
More informationAxiomatic set theory. Chapter Why axiomatic set theory?
Chapter 1 Axiomatic set theory 1.1 Why axiomatic set theory? Essentially all mathematical theories deal with sets in one way or another. In most cases, however, the use of set theory is limited to its
More informationThe roots of computability theory. September 5, 2016
The roots of computability theory September 5, 2016 Algorithms An algorithm for a task or problem is a procedure that, if followed step by step and without any ingenuity, leads to the desired result/solution.
More informationWhat are Axioms of Set Theory?
Set-theorists use the term Axiom (of Set Theory) quite freely. What do they mean by it? Examples Axioms of ZFC: Axiom of Extensionality Pairing Axiom Separation Axiom Union Axiom Powerset Axiom Axiom of
More informationCONCEPT CALCULUS by Harvey M. Friedman September 5, 2013 GHENT
1 CONCEPT CALCULUS by Harvey M. Friedman September 5, 2013 GHENT 1. Introduction. 2. Core system, CORE. 3. Equivalence system, EQ. 4. First extension system, EX1. 5. Second extension system, EX2. 6. Third
More informationAlmost von Neumann, Definitely Gödel: The Second Incompleteness Theorem s Early Story
L&PS Logic and Philosophy of Science Vol. IX, No. 1, 2011, pp. 151-158 Almost von Neumann, Definitely Gödel: The Second Incompleteness Theorem s Early Story Giambattista Formica Dipartimento di Filosofia,
More informationPhilosophies of Mathematics. The Search for Foundations
Philosophies of Mathematics The Search for Foundations Foundations What are the bedrock, absolutely certain, immutable truths upon which mathematics can be built? At one time, it was Euclidean Geometry.
More information... The Sequel. a.k.a. Gödel's Girdle
... The Sequel a.k.a. Gödel's Girdle Formal Systems A Formal System for a mathematical theory consists of: 1. A complete list of the symbols to be used. 2. Rules of syntax The rules that determine properly
More informationTowards a Contemporary Ontology
Towards a Contemporary Ontology The New Dual Paradigm in Natural Sciences: Part I Module 2 Class 3: The issue of the foundations of mathematics Course WI-FI-BASTI1 2014/15 Introduction Class 3: The issue
More informationDiscrete Mathematics
Discrete Mathematics Yi Li Software School Fudan University March 13, 2017 Yi Li (Fudan University) Discrete Mathematics March 13, 2017 1 / 1 Review of Lattice Ideal Special Lattice Boolean Algebra Yi
More informationThe Philosophy of Mathematics after Foundationalism
The Philosophy of Mathematics after Foundationalism Dan Goodman November 5, 2005 1 Introduction Philosophy of maths has come to mean something quite specific; namely logic, set theory and the foundations
More informationAn Intuitively Complete Analysis of Gödel s Incompleteness
An Intuitively Complete Analysis of Gödel s Incompleteness JASON W. STEINMETZ (Self-funded) A detailed and rigorous analysis of Gödel s proof of his first incompleteness theorem is presented. The purpose
More informationTheory of Computation CS3102 Spring 2014
Theory of Computation CS0 Spring 0 A tale of computers, math, problem solving, life, love and tragic death Nathan Brunelle Department of Computer Science University of Virginia www.cs.virginia.edu/~njbb/theory
More informationThe multiverse perspective on determinateness in set theory
The multiverse perspective on determinateness in set theory Joel David Hamkins New York University, Philosophy The City University of New York, Mathematics College of Staten Island of CUNY The CUNY Graduate
More informationWHY ISN T THE CONTINUUM PROBLEM ON THE MILLENNIUM ($1,000,000) PRIZE LIST?
WHY ISN T THE CONTINUUM PROBLEM ON THE MILLENNIUM ($1,000,000) PRIZE LIST? Solomon Feferman CSLI Workshop on Logic, Rationality and Intelligent Interaction Stanford, June 1, 2013 Why isn t the Continuum
More informationGeneralizing Gödel s Constructible Universe:
Generalizing Gödel s Constructible Universe: Ultimate L W. Hugh Woodin Harvard University IMS Graduate Summer School in Logic June 2018 Ordinals: the transfinite numbers is the smallest ordinal: this is
More informationModel Theory of Second Order Logic
Lecture 2 1, 2 1 Department of Mathematics and Statistics University of Helsinki 2 ILLC University of Amsterdam March 2011 Outline Second order characterizable structures 1 Second order characterizable
More informationThe constructible universe
The constructible universe In this set of notes I want to sketch Gödel s proof that CH is consistent with the other axioms of set theory. Gödel s argument goes well beyond this result; his identification
More informationParadox Machines. Christian Skalka The University of Vermont
Paradox Machines Christian Skalka The University of Vermont Source of Mathematics Where do the laws of mathematics come from? The set of known mathematical laws has evolved over time (has a history), due
More informationHuan Long Shanghai Jiao Tong University
Huan Long Shanghai Jiao Tong University Paradox Equinumerosity Cardinal Numbers Infinite Cardinals Paradox and ZFC Equinumerosity Ordering Countable sets Paradox Axiomatic set theory Modern set theory
More informationWe begin with a standard definition from model theory.
1 IMPOSSIBLE COUNTING by Harvey M. Friedman Distinguished University Professor of Mathematics, Philosophy, and Computer Science Emeritus Ohio State University Columbus, Ohio 43235 June 2, 2015 DRAFT 1.
More informationThe hierarchy of second-order set theories between GBC and KM and beyond
The hierarchy of second-order set theories between GBC and KM and beyond Joel David Hamkins City University of New York CUNY Graduate Center Mathematics, Philosophy, Computer Science College of Staten
More informationTrees and generic absoluteness
in ZFC University of California, Irvine Logic in Southern California University of California, Los Angeles November 6, 03 in ZFC Forcing and generic absoluteness Trees and branches The method of forcing
More informationIndefiniteness, definiteness and semi-intuitionistic theories of sets
Indefiniteness, definiteness and semi-intuitionistic theories of sets Michael Rathjen Department of Pure Mathematics University of Leeds Brouwer Symposium Amsterdam December 9th, 2016 The continuum hypothesis,
More informationSet-theoretic potentialism and the universal finite set
Set-theoretic potentialism and the universal finite set Joel David Hamkins Oxford University University College, Oxford & City University of New York CUNY Graduate Center College of Staten Island Scandivavian
More informationInterpreting classical theories in constructive ones
Interpreting classical theories in constructive ones Jeremy Avigad Department of Philosophy Carnegie Mellon University avigad+@cmu.edu http://macduff.andrew.cmu.edu 1 A brief history of proof theory Before
More informationThe Search for the Perfect Language
The Search for the Perfect Language I'll tell you how the search for certainty led to incompleteness, uncomputability & randomness, and the unexpected result of the search for the perfect language. Bibliography
More informationThe logic of Σ formulas
The logic of Σ formulas Andre Kornell UC Davis BLAST August 10, 2018 Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, 2018 1 / 22 the Vienna Circle The meaning of a proposition is the
More informationBetween proof theory and model theory Three traditions in logic: Syntactic (formal deduction)
Overview Between proof theory and model theory Three traditions in logic: Syntactic (formal deduction) Jeremy Avigad Department of Philosophy Carnegie Mellon University avigad@cmu.edu http://andrew.cmu.edu/
More informationExtremely large cardinals in the absence of Choice
Extremely large cardinals in the absence of Choice David Asperó University of East Anglia UEA pure math seminar, 8 Dec 2014 The language First order language of set theory. Only non logical symbol: 2 The
More informationRussell s logicism. Jeff Speaks. September 26, 2007
Russell s logicism Jeff Speaks September 26, 2007 1 Russell s definition of number............................ 2 2 The idea of reducing one theory to another.................... 4 2.1 Axioms and theories.............................
More informationSet Theory: Forcing and Semantics. Roger Bishop Jones
Set Theory: Forcing and Semantics Roger Bishop Jones Contents Preface 2 1 Introduction 2 2 Semantic Logicism 3 2.1 formalism........................ 4 2.2 Some Notes on Carnap................. 4 3 Forcing
More informationCantor and sets: La diagonale du fou
Judicaël Courant 2011-06-17 Lycée du Parc (moving to Lycée La Martinière-Monplaisir) Outline 1 Cantor s paradise 1.1 Introduction 1.2 Countable sets 1.3 R is not countable 1.4 Comparing sets 1.5 Cardinals
More informationDEDUCTIVE PLURALISM 1
1 2 3 4 5 6 7 8 9 10 11 12 13 DEDUCTIVE PLURALISM 1 John Hosack ABSTRACT This paper proposes an approach to the philosophy of mathematics, deductive pluralism, that is designed to satisfy the criteria
More informationBootstrapping Mathematics
Bootstrapping Mathematics Masahiko Sato Graduate School of Informatics, Kyoto University Mathematical Logic: Development and Evolution into Various Sciences Kanazawa, Japan March 9, 2012 Contents What
More informationUnsolvable problems, the Continuum Hypothesis, and the nature of infinity
Unsolvable problems, the Continuum Hypothesis, and the nature of infinity W. Hugh Woodin Harvard University January 9, 2017 V : The Universe of Sets The power set Suppose X is a set. The powerset of X
More informationMinimal models of second-order set theories
Minimal models of second-order set theories Kameryn J Williams CUNY Graduate Center Set Theory Day 2016 March 11 K Williams (CUNY) Minimal models of second-order set theories 2016 March 11 1 / 17 A classical
More informationReview: Stephen G. Simpson (1999) Subsystems of Second-Order Arithmetic (Springer)
Review: Stephen G. Simpson (1999) Subsystems of Second-Order Arithmetic (Springer) Jeffrey Ketland, February 4, 2000 During the nineteenth century, and up until around 1939, many major mathematicians were
More informationLecture 14 Rosser s Theorem, the length of proofs, Robinson s Arithmetic, and Church s theorem. Michael Beeson
Lecture 14 Rosser s Theorem, the length of proofs, Robinson s Arithmetic, and Church s theorem Michael Beeson The hypotheses needed to prove incompleteness The question immediate arises whether the incompleteness
More informationA Height-Potentialist View of Set Theory
A Height-Potentialist View of Set Theory Geoffrey Hellman September, 2015 SoTFoM Vienna 1 The Modal-Structural Framework Background logic: S5 quantified modal logic with secondorder or plurals logic, without
More informationSet Theory History. Martin Bunder. September 2015
Set Theory History Martin Bunder September 2015 What is a set? Possible Definition A set is a collection of elements having a common property Abstraction Axiom If a(x) is a property ( y)( x)(x y a(x))
More informationAutomata Theory. Definition. Computational Complexity Theory. Computability Theory
Outline THEORY OF COMPUTATION CS363, SJTU What is Theory of Computation? History of Computation Branches and Development Xiaofeng Gao Dept. of Computer Science Shanghai Jiao Tong University 2 The Essential
More informationProjective well-orderings of the reals and forcing axioms
Projective well-orderings of the reals and forcing axioms Andrés Eduardo Department of Mathematics Boise State University 2011 North American Annual Meeting UC Berkeley, March 24 27, 2011 This is joint
More informationThomas Jech. The Pennsylvania State University. Until the 19th century the study of the innite was an exclusive domain of
THE INFINITE Thomas Jech The Pennsylvania State University Until the 19th century the study of the innite was an exclusive domain of philosophers and theologians. For mathematicians, while the concept
More informationINTRODUCTION TO CARDINAL NUMBERS
INTRODUCTION TO CARDINAL NUMBERS TOM CUCHTA 1. Introduction This paper was written as a final project for the 2013 Summer Session of Mathematical Logic 1 at Missouri S&T. We intend to present a short discussion
More informationThe modal logic of forcing
Joel David Hamkins New York University, Philosophy The City University of New York, Mathematics College of Staten Island of CUNY The CUNY Graduate Center London, August 5 6, 2011 This is joint work with
More informationOuter Model Satisfiability. M.C. (Mack) Stanley San Jose State
Outer Model Satisfiability M.C. (Mack) Stanley San Jose State The Universe of Pure Sets V 0 = V α+1 = P(V α ) = { x : x V α } V λ = V α, λ a limit α
More informationTallness and Level by Level Equivalence and Inequivalence
Tallness and Level by Level Equivalence and Inequivalence Arthur W. Apter Department of Mathematics Baruch College of CUNY New York, New York 10010 USA and The CUNY Graduate Center, Mathematics 365 Fifth
More information185.A09 Advanced Mathematical Logic
185.A09 Advanced Mathematical Logic www.volny.cz/behounek/logic/teaching/mathlog13 Libor Běhounek, behounek@cs.cas.cz Lecture #1, October 15, 2013 Organizational matters Study materials will be posted
More informationThe Reflection Theorem
The Reflection Theorem Formalizing Meta-Theoretic Reasoning Lawrence C. Paulson Computer Laboratory Lecture Overview Motivation for the Reflection Theorem Proving the Theorem in Isabelle Applying the Reflection
More informationIntroduction to Logic and Axiomatic Set Theory
Introduction to Logic and Axiomatic Set Theory 1 Introduction In mathematics, we seek absolute rigor in our arguments, and a solid foundation for all of the structures we consider. Here, we will see some
More informationOctober 12, Complexity and Absoluteness in L ω1,ω. John T. Baldwin. Measuring complexity. Complexity of. concepts. to first order.
October 12, 2010 Sacks Dicta... the central notions of model theory are absolute absoluteness, unlike cardinality, is a logical concept. That is why model theory does not founder on that rock of undecidability,
More informationGödel and Formalism freeness. Juliette Kennedy
Gödel and Formalism freeness Juliette Kennedy Tait, The 5 Questions (2008) Logic Another line of thought has to do with more logical considerations, consisting of a collection of ideas associated with
More informationPoincaré s Thesis CONTENTS. Peter Fekete
Poincaré s Thesis CONTENTS Peter Fekete COPYWRITE PETER FEKETE 6 TH OCT. 2011 NO PART OF THIS PAPER MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY,
More information2.2 Lowenheim-Skolem-Tarski theorems
Logic SEP: Day 1 July 15, 2013 1 Some references Syllabus: http://www.math.wisc.edu/graduate/guide-qe Previous years qualifying exams: http://www.math.wisc.edu/ miller/old/qual/index.html Miller s Moore
More informationContradictions in mathematics
Contradictions in mathematics Manuel Bremer Philosophisches Institut, Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, 40225 Düsseldorf, Germany E-mail: bremer@mbph.de Contradictions are typically
More informationCreative Objectivism, a powerful alternative to Constructivism
Creative Objectivism, a powerful alternative to Constructivism Copyright c 2002 Paul P. Budnik Jr. Mountain Math Software All rights reserved Abstract It is problematic to allow reasoning about infinite
More informationThis section will take the very naive point of view that a set is a collection of objects, the collection being regarded as a single object.
1.10. BASICS CONCEPTS OF SET THEORY 193 1.10 Basics Concepts of Set Theory Having learned some fundamental notions of logic, it is now a good place before proceeding to more interesting things, such as
More informationCSCI3390-Lecture 6: An Undecidable Problem
CSCI3390-Lecture 6: An Undecidable Problem September 21, 2018 1 Summary The language L T M recognized by the universal Turing machine is not decidable. Thus there is no algorithm that determines, yes or
More informationPROOF-THEORETIC REDUCTION AS A PHILOSOPHER S TOOL
THOMAS HOFWEBER PROOF-THEORETIC REDUCTION AS A PHILOSOPHER S TOOL 1. PROOF-THEORETIC REDUCTION AND HILBERT S PROGRAM Hilbert s program in the philosophy of mathematics comes in two parts. One part is a
More informationcse371/mat371 LOGIC Professor Anita Wasilewska Fall 2018
cse371/mat371 LOGIC Professor Anita Wasilewska Fall 2018 Chapter 7 Introduction to Intuitionistic and Modal Logics CHAPTER 7 SLIDES Slides Set 1 Chapter 7 Introduction to Intuitionistic and Modal Logics
More informationGeometry I (CM122A, 5CCM122B, 4CCM122A)
Geometry I (CM122A, 5CCM122B, 4CCM122A) Lecturer: Giuseppe Tinaglia Office: S5.31 Office Hours: Wed 1-3 or by appointment. E-mail: giuseppe.tinaglia@kcl.ac.uk Course webpage: http://www.mth.kcl.ac.uk/
More informationForcing axioms and inner models
Outline Andrés E. Department of Mathematics California Institute of Technology Boise State University, February 27 2008 Outline Outline 1 Introduction 2 Inner models M with ω1 M = ω 1 Inner models M with
More informationThe Calculus of Inductive Constructions
The Calculus of Inductive Constructions Hugo Herbelin 10th Oregon Programming Languages Summer School Eugene, Oregon, June 16-July 1, 2011 1 Outline - A bit of history, leading to the Calculus of Inductive
More informationLCF + Logical Frameworks = Isabelle (25 Years Later)
LCF + Logical Frameworks = Isabelle (25 Years Later) Lawrence C. Paulson, Computer Laboratory, University of Cambridge 16 April 2012 Milner Symposium, Edinburgh 1979 Edinburgh LCF: From the Preface the
More informationProducts, Relations and Functions
Products, Relations and Functions For a variety of reasons, in this course it will be useful to modify a few of the settheoretic preliminaries in the first chapter of Munkres. The discussion below explains
More informationThe Legacy of Hilbert, Gödel, Gentzen and Turing
The Legacy of Hilbert, Gödel, Gentzen and Turing Amílcar Sernadas Departamento de Matemática - Instituto Superior Técnico Security and Quantum Information Group - Instituto de Telecomunicações TULisbon
More informationGödel s Proof. Henrik Jeldtoft Jensen Dept. of Mathematics Imperial College. Kurt Gödel
Gödel s Proof Henrik Jeldtoft Jensen Dept. of Mathematics Imperial College Kurt Gödel 24.4.06-14.1.78 1 ON FORMALLY UNDECIDABLE PROPOSITIONS OF PRINCIPIA MATHEMATICA AND RELATED SYSTEMS 11 by Kurt Gödel,
More informationThe Gödel Hierarchy and Reverse Mathematics
The Gödel Hierarchy and Reverse Mathematics Stephen G. Simpson Pennsylvania State University http://www.math.psu.edu/simpson/ simpson@math.psu.edu Symposium on Hilbert s Problems Today Pisa, Italy April
More informationThrowing Darts, Time, and the Infinite
Erkenn DOI 10.1007/s10670-012-9371-x ORIGINAL PAPER Throwing Darts, Time, and the Infinite Jeremy Gwiazda Received: 23 August 2011 / Accepted: 2 March 2012 Ó Springer Science+Business Media B.V. 2012 Abstract
More informationKrivine s Intuitionistic Proof of Classical Completeness (for countable languages)
Krivine s Intuitionistic Proof of Classical Completeness (for countable languages) Berardi Stefano Valentini Silvio Dip. Informatica Dip. Mat. Pura ed Applicata Univ. Torino Univ. Padova c.so Svizzera
More informationResearch Statement. Peter Koellner
Research Statement Peter Koellner My research concerns the search for and justification of new axioms in mathematics. The need for new axioms arises from the independence results. Let me explain. In reasoning
More informationRecursion Theory. Joost J. Joosten
Recursion Theory Joost J. Joosten Institute for Logic Language and Computation University of Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam Room P 3.26, +31 20 5256095 jjoosten@phil.uu.nl www.phil.uu.nl/
More informationGÖDEL S COMPLETENESS AND INCOMPLETENESS THEOREMS. Contents 1. Introduction Gödel s Completeness Theorem
GÖDEL S COMPLETENESS AND INCOMPLETENESS THEOREMS BEN CHAIKEN Abstract. This paper will discuss the completeness and incompleteness theorems of Kurt Gödel. These theorems have a profound impact on the philosophical
More informationTRUTH TELLERS. Volker Halbach. Scandinavian Logic Symposium. Tampere
TRUTH TELLERS Volker Halbach Scandinavian Logic Symposium Tampere 25th August 2014 I m wrote two papers with Albert Visser on this and related topics: Self-Reference in Arithmetic, http://www.phil.uu.nl/preprints/lgps/number/316
More informationFoundations in Mathematics: Modern Views
Conference Foundations in Mathematics: Modern Views Booklet of Abstracts LMU Munich, 4 7 April 2018 Geschwister-Scholl-Platz 1, 80539 München, Germany Organizing committee Vera Gahlen, Levin Hornischer,
More information1 FUNDAMENTALS OF LOGIC NO.1 WHAT IS LOGIC Tatsuya Hagino hagino@sfc.keio.ac.jp lecture URL https://vu5.sfc.keio.ac.jp/slide/ 2 Course Summary What is the correct deduction? Since A, therefore B. It is
More informationSOME TRANSFINITE INDUCTION DEDUCTIONS
SOME TRANSFINITE INDUCTION DEDUCTIONS SYLVIA DURIAN Abstract. This paper develops the ordinal numbers and transfinite induction, then demonstrates some interesting applications of transfinite induction.
More informationA simple maximality principle
A simple maximality principle arxiv:math/0009240v1 [math.lo] 28 Sep 2000 Joel David Hamkins The City University of New York Carnegie Mellon University http://math.gc.cuny.edu/faculty/hamkins February 1,
More informationArgument. whenever all the assumptions are true, then the conclusion is true. If today is Wednesday, then yesterday is Tuesday. Today is Wednesday.
Logic and Proof Argument An argument is a sequence of statements. All statements but the first one are called assumptions or hypothesis. The final statement is called the conclusion. An argument is valid
More informationHandbook of Logic and Proof Techniques for Computer Science
Steven G. Krantz Handbook of Logic and Proof Techniques for Computer Science With 16 Figures BIRKHAUSER SPRINGER BOSTON * NEW YORK Preface xvii 1 Notation and First-Order Logic 1 1.1 The Use of Connectives
More informationGödel in class. Achim Feldmeier Brno - Oct 2010
Gödel in class Achim Feldmeier Brno - Oct 2010 Philosophy lost key competence to specialized disciplines: right life (happyness, morals) Christianity science and technology Natural Sciences social issues
More informationKRIPKE S THEORY OF TRUTH 1. INTRODUCTION
KRIPKE S THEORY OF TRUTH RICHARD G HECK, JR 1. INTRODUCTION The purpose of this note is to give a simple, easily accessible proof of the existence of the minimal fixed point, and of various maximal fixed
More informationChapter 2: Introduction to Propositional Logic
Chapter 2: Introduction to Propositional Logic PART ONE: History and Motivation Origins: Stoic school of philosophy (3rd century B.C.), with the most eminent representative was Chryssipus. Modern Origins:
More informationNonclassical logics (Nichtklassische Logiken)
Nonclassical logics (Nichtklassische Logiken) VU 185.249 (lecture + exercises) http://www.logic.at/lvas/ncl/ Chris Fermüller Technische Universität Wien www.logic.at/people/chrisf/ chrisf@logic.at Winter
More informationA BRIEF INTRODUCTION TO ZFC. Contents. 1. Motivation and Russel s Paradox
A BRIEF INTRODUCTION TO ZFC CHRISTOPHER WILSON Abstract. We present a basic axiomatic development of Zermelo-Fraenkel and Choice set theory, commonly abbreviated ZFC. This paper is aimed in particular
More informationPropositional and Predicate Logic - XIII
Propositional and Predicate Logic - XIII Petr Gregor KTIML MFF UK WS 2016/2017 Petr Gregor (KTIML MFF UK) Propositional and Predicate Logic - XIII WS 2016/2017 1 / 22 Undecidability Introduction Recursive
More information